Denise: Deep Robust Principal Component Analysis for Positive Semidefinite Matrices

• DOI: 10.48550/arxiv.2004.13612
• Publication Date: 2020-01-01
• Authors: Herrera, Calypso; Krach, Florian; i...; Kratsios, Anastas...; Ruyssen, Pierre; Teichmann, Josef
• Keywords: FOS: Computer and information sciences; Computer Science - Machine Learning; Low-rank plus sparse decomposition; positive semidefinite matrices; deep neural networks; geometric deep learning; universal approximation; universal approximation; positive semidefinite matrices; Machine Learning (stat.ML); Computational Finance (q-fin.CP); 02 engineering and technology; Machine Learning (cs.LG); Low-rank plus sparse decomposition; FOS: Economics and business; Quantitative Finance - Computational Finance; deep neural networks; geometric deep learning; Statistics - Machine Learning; Optimization and Control (math.OC); 0202 electrical engineering, electronic engineering, information engineering; FOS: Mathematics; Mathematics - Optimization and Control

ISSN:2835-8856<br/>Transactions on Machine Learning Research<br/>The robust PCA of covariance matrices plays an essential role when isolating key explanatory features. The currently available methods for performing such a low-rank plus sparse decomposition are matrix specific, meaning, those algorithms must re-run for every new matrix. Since these algorithms are computationally expensive, it is preferable to learn and store a function that nearly instantaneously performs this decomposition when evaluated. Therefore, we introduce Denise, a deep learning-based algorithm for robust PCA of covariance matrices, or more generally, of symmetric positive semidefinite matrices, which learns precisely such a function. Theoretical guarantees for Denise are provided. These include a novel universal approximation theorem adapted to our geometric deep learning problem and convergence to an optimal solution to the learning problem. Our experiments show that Denise matches state-of-the-art performance in terms of decomposition quality, while being approximately 2000x faster than the state-of-the-art, principal component pursuit (PCP), and 2000x faster than the current speed-optimized method, fast PCP.<br/>